As pointed out in [1]}, requesting that the solutions of the CH equation vanishes on some open set is a very strong condition. A question that the author have been asking himself since the work [2]} is: Would it be possible to relax such a strong condition? The answer is positive and is given by Theorem REF  for the system (REF ), and Theorem REF  for the CH equation in particular. In these theorems we relaxed the hypothesis that \(u\)  vanish on some open set by the assumption that there exists an open set such that the solution \(u\)  depends on \(t\) . If this happens, then the condition (REF ) will again imply that \(f_t(x)\)  vanishes and the conservation of energy once more implies that the solution vanishes everywhere. Furthermore, this fact can be used to give a demonstration that the solutions of the CH equation cannot be compactly supported if the initial data is non-trivial. Such demonstration is rather different from those given in [3]}, [4]}, [5]} and, moreover, can be applied to periodic problems.
